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We usually write co(n; P)=co(n), ~2(n; P ) = ~ ( n ) . In a previous paper [37], we obtained sharp inequalities for the frequencies of large deviations of co(n; E) and ~(n; E) from their normal order of magnitude. Those inequalities included refinements of a special case of a general theorem due to Elliott [11, Theorem 6] concerning large deviations of/(g(n)), where / is a strongly additive arithmetic function and g(n) is a positive-valued polynomial in n with integral coefficients. Elliott 's result was in turn a refinement (under stronger hypotheses) of a theorem of U~davinis [55]. (The result of U~davinis is stated as Theorem 3.3 in Kubilius [28].) The methods used in [37] were "almost" elementary. Here we shall use more difficult methods to obtain asymptotic formulas for large deviations of co(n; E) and ~(n; E). We shall also generalize some of the results of [37] and give some applications. For a partial survey of the literature in this area, see [39]. In order to state our main theorems, it is necessary to introduce further notation which will be used throughout this paper. First, we define |
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